In this appendix, we aim to prove that two giant atoms with decoherence-free indirect coupling cannot exhibit nonreciprocal transfer even in the presence of a direct coupling between them. As shown in Fig. REF , both emitters \(a\)  and \(b\)  couple with the waveguide twice in the braided manner, i.e., emitter \(a\)  (\(b\) ) couples with the waveguide at \(x=x_{1}\)  and \(x=x_{3}\)  (\(x=x_{2}\)  and \(x=x_{4}\) ). The emitter-waveguide couplings are assumed to be identical (i.e., \(g\) ) and the coupling ports are evenly spaced by distance \(d\) . Such a braided structure can be implemented, for example, with a S-type waveguide [1]}, [2]}. To distinguish from the small-atom dimer in Fig. REF (a), we refer to this model as the giant-atom dimer. Once again, we consider the direct coupling \(|J|e^{\pm i\theta }\)  between \(a\)  and \(b\) . In this case, the effective equations of the probability amplitudes \(c_{a}\)  and \(c_{b}\)  are written as
\(\begin{split}&\frac{dc_{a}(t)}{dt}=-\gamma c_{a}(t)-\frac{\gamma }{2}(3D_{1,b}+D_{3,b})\\&\qquad \qquad -\gamma D_{2,a}-i|J|e^{i\theta }c_{b}(t),\\&\frac{dc_{b}(t)}{dt}=-\gamma c_{b}(t)-\frac{\gamma }{2}(3D_{1,a}+D_{3,a})\\&\qquad \qquad -\gamma D_{2,b}-i|J|e^{-i\theta }c_{a}(t),\end{split}\) 
